Microbundle
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1 Introduction
The concept of a microbundle of dimension was first introduced in [Milnor1964] to give a model for the tangent bundle of an n-dimensional topological manifold. Later [Kister1964] showed that every microbundle uniquely determines a topological
-bundle.
Definition 1.1 [Milnor1964] .
An![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![(E,B,i,j)](/images/math/7/1/0/71015335477690380a3d9e5627f000db.png)
![\displaystyle B\xrightarrow{i} E\xrightarrow{j} B](/images/math/4/0/a/40a68b194c05ed02a16adc8c005f25f3.png)
- for all
there exist open neigbourhood
, an open neighbourhood
of
and a homeomorphism
which makes the following diagram commute:
![\displaystyle \xymatrix{& V \ar[dr]^{j|_V} \ar[dd]^h \\ U \ar[dr]_{\times 0} \ar[ur]^{i|_U} & & U \\ & U \times \Rr^n \ar[ur]_{p_1}}](/images/math/a/3/8/a38e57a84d0d2eb3609b3f55e4fe1d0f.png)
For any space define the diagonal embedding
![\displaystyle \Delta_M \colon M \to M \times M;x \mapsto (x,x)~.](/images/math/2/8/e/28ecf11ae15ce5079a805063ce8dc025.png)
If is a differentiable
-manifold the normal bundle of
is the tangent bundle
of
.
In the topological category we have:
Example 1.2 [Milnor1964, Lemma 2.1].
Let be topological
-manifold, and let
be the projection onto the first factor. Then
![\displaystyle (M \times M, M, \Delta_M, p_1)](/images/math/0/8/f/08febb705b78edb17c9c3a6116f9b713.png)
is an -dimensional microbundle, the tangent microbundle
of
.
![\pi \colon E \to B](/images/math/9/a/5/9a5efa4effb7d97096da6951dd535385.png)
![\Rr^n](/images/math/a/f/5/af5902cb024e9f035b1f86e86b207a38.png)
![s \colon B \to E](/images/math/8/1/8/81843d6c6d5d46d9c73a0ae35765141c.png)
![\displaystyle (E, B, s, \pi)](/images/math/8/d/e/8de96595ed823c69d3a289b186d86d41.png)
is an -dimensional microbundle.
Definition 1.4.
Two microbundles ,
over the same space
are isomorphic if there exist neighbourhoods
of
and
of
and a homeomorphism
making the following diagram commute:
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Theorem 1.5 [Kister1964, Theorem 2].
Let be an
-dimensional microbundle. Then there is a neighbourhood of
,
such that:
-
is the total space of a topological
-bundle over
.
- The inclusion
is a microbundle isomorphism
- If
is any other such neighbourhood of
then there is a
-bundle isomorphism
.
2 References
- [Kister1964] J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR0180986 (31 #5216) Zbl 0131.20602
- [Milnor1964] J. Milnor, Microbundles. I, Topology 3 (1964), no.suppl. 1, 53–80. MR0161346 (28 #4553b) Zbl 0124.38404